You are right and my apologies for contributing to the derailing. I'm increasingly convinced he/she is trolling as some of these things are so totally off-the-wall it seems a bit too outlandishly ignorant to be real, even for the interwebs...
I think having certain meeting locations (at varying times) is good, particularly at the grad-student focused events (e.g. APAGS events). You could rely on a network model (i.e. make extras and give everyone a handful to give their friends). If enough people are on board and willing, you could even set up a rotating schedule of folks roaming near registration desks. I'm not familiar with the Orlando location, but there are always more heavily trafficked areas at these things.
Of course, you could always walk around wearing a sandwich board that says "Ask me why APA hates graduate students"...but THAT one might get you in trouble
Edit: Depending on whether the buttons themselves are "Protest" or "Show of support" would APAGS be willing to stand by this as an important issue and perhaps make a case for formal involvement in distribution? Any chance these being kept AT the registration desks or distributed with student registration packets? Figure that is a huge long-shot and APA is not exactly known for promoting open and honest communication on professional issues, but just brainstorming....